Prepositions (in/near/throughout/with reference to) often begin sentences.
"Mine" is a possessive pronoun, however, and I don't know of any rules about starting a sentence with one or not.
In reference to some of the comments in posts above:
I think "mine and Jenny's" sounds right because it would be OK in these sentences: The car is mine and Jenny's.
or simply It's mine and Jenny's.
The Harbrace College Handbook
partly clears up our original problem with the dual ownership of that car by explaining when to use apostrophes: "To indicate individual ownership, add the apostrophe and s to each name:
Joan's and Sam's apartments (each own individual apartments)
Joe and Betty's mail (indicating mail belonging jointly to Joe and Betty)."
But the other confusing part of the infamous "It's my and Jenny's car
" sentence is the choice of either my or mine
. "My" is an adjective and modifies
the word "car" (It's my car.) "Mine" is a pronoun and replaces
the word "car" (It's mine.)
So, it is my opinion that the correct form of the sentence is "It's my and Jenny's car"
but after spending so much time looking in grammar books trying to explain why
I will be sure never to use this sentence again in my life.